Global existence for the einstein vacuum equations in wave coordinates

Hans Lindblad, Igor Rodnianski

Research output: Contribution to journalArticle

80 Citations (Scopus)

Abstract

We prove global stability of Minkowski space for the Einstein vacuum equations in harmonic (wave) coordinate gauge for the set of restricted data coinciding with the Schwarzschild solution in the neighborhood of space-like infinity. The result contradicts previous beliefs that wave coordinates are "unstable in the large" and provides an alternative approach to the stability problem originally solved ( for unrestricted data, in a different gauge and with a precise description of the asymptotic behavior at null infinity) by D. Christodoulou and S. Klainerman. Using the wave coordinate gauge we recast the Einstein equations as a system of quasilinear wave equations and, in absence of the classical null condition, establish a small data global existence result. In our previous work we introduced the notion of a weak null condition and showed that the Einstein equations in harmonic coordinates satisfy this condition.The result of this paper relies on this observation and combines it with the vector field method based on the symmetries of the standard Minkowski space. In a forthcoming paper we will address the question of stability of Minkowski space for the Einstein vacuum equations in wave coordinates for all "small" asymptotically flat data and the case of the Einstein equations coupled to a scalar field.

Original languageEnglish (US)
Pages (from-to)43-110
Number of pages68
JournalCommunications In Mathematical Physics
Volume256
Issue number1
DOIs
StatePublished - May 1 2005

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Global Existence
Albert Einstein
Minkowski Space
Vacuum
Einstein Equations
Null Condition
Minkowski space
Einstein equations
Gauge
vacuum
Harmonic
infinity
Infinity
Quasilinear Wave Equation
harmonics
Global Stability
Scalar Field
Existence Results
Null
Vector Field

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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Global existence for the einstein vacuum equations in wave coordinates. / Lindblad, Hans; Rodnianski, Igor.

In: Communications In Mathematical Physics, Vol. 256, No. 1, 01.05.2005, p. 43-110.

Research output: Contribution to journalArticle

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